A Flaw of General Relativity, a New Metric and Cosmological Implications [Technical]

planck2

oh i wouldn't say that at all. I am perfectly capable of finding the flaw in an arguement. Your's being a prime example

Zanket

planck2 wrote:

I am perfectly capable of finding the flaw in an arguement. Your's being a prime example

What post # of yours contained a flaw of my paper that I did not subsequently refute? Empty/incomplete claims don't count.

planck2

take this for example, you claim your metric gives approximation to light bending effects and agrees with experimental measurements. I have not seen your calculation. You cannot do a single calculation yourself and rely on others to do them and you shoot them down when they say something which you claim you can't even verify. You talk nonsense. You are wasting my time. If I were the mod I would have finished this discussion ages ago.

breadmonkey

planck2 wrote:

take this for example, you claim your metric gives approximation to light bending effects and agrees with experimental measurements. I have not seen your calculation. You cannot do a single calculation yourself and rely on others to do them and you shoot them down when they say something which you claim you can't even verify. You talk nonsense. You are wasting my time. If I were the mod I would have finished this discussion ages ago.

Good job you aren't a mod then.

planck2

Well I don't about that. I will listen to well reasoned arguments and people to are willing to discuss, but not those who consistently claim to be correct when they are not

Zanket

planck2 wrote:

take this for example, you claim your metric gives approximation to light bending effects and agrees with experimental measurements. I have not seen your calculation.

There’s already a reader comment in section 6 that covers this:

Reader: No experiments are listed.

Author: For the experiment referenced and all those in weaker gravity (i.e. all other experimental tests of the Schwarzschild metric to date), this section shows that the new metric and the Schwarzschild metric make the same predictions. Then, for example, the new metric predicts that the relativistic orbital precession of Mercury is 42.98 arc seconds per century, the same as the Schwarzschild metric predicts.

Do you think it is impossible to show with logic alone that my metric will agree with the Schwarzschild metric to all significant digits for all experimental tests to date? Why are you insistent that predictions for specific experiments must be calculated? Only one experiment is needed: the one with the strongest gravity to date. That’s the one that section 6 references.

You cannot do a single calculation yourself and rely on others to do them ...

You’re the pot calling the kettle black. R_ab != 0 is your claim. If you want me to do the calculation for that, then you are relying on me to do it for you. Anyone can post an equation here. Unless you show your work you have not supported your claim. (I did the calculation for my metric and it shows that R_ab = 0. See, by your own logic I just proved you wrong. So what are you complaining about?)

... and you shoot them down when they say something which you claim you can't even verify. You talk nonsense.

I am not obligated to simply trust you when I can’t verify parts of your incomplete claim. For example, if there’s an experiment of natural phenomena that tests Ricci curvature, then list it here. How hard could that be, if you know about such an experiment? You should have had the experiment in mind before you ever made the claim. If you refuse to list it, and I can’t verify it despite an extensive search, then of course I’m going to be unconvinced. Any reasonable person would think the same.

If I were the mod I would have finished this discussion ages ago.

Censorship does seem to be back in favor these days.

I will listen to well reasoned arguments and people to are willing to discuss, but not those who consistently claim to be correct when they are not

Your insistence that logic alone cannot prove anything shows that you are not willing to listen to a well-reasoned argument.

If you think the paper is wrong, then by all means quote some part of it and make a supported claim that refutes it. I don’t just claim to be correct, I show it, whereas you do not support your claims. I am willing to discuss any claim you can make that has a basis. Every time someone has quotes the paper here or anywhere and says what they think is wrong with it, I give my reasons to refute that, unless I can’t, in which case I change the paper if I can save it. Whereas I give my bases and argue my points to the last side standing, you make empty claims and want to have the thread locked if I don’t take them on faith alone. Who’s being scientific here?

Professor_Fink

Ok, this ends now. Here is a derivation for R_00 for both Zankets metric and the Schwarzschild metric. Note that it took 71 equations to derive this one element of the tensor. I do not plan on repeating this another 15 times.

Clearly R_00 is nonzero for Zankets metric, and R_00=0 for the Schwarzschild and Minkowski metrics.

This means that Zankets metric is not conformal to flat space, which means he cannot use arguements based on special relativity with his metric and remain consistent. Additionally R_ab is always observed to be 0 when T_ab=0 otherwise special relativity would essentially be wrong. Muon decay experiments, etc., all verify special relativity, so Zanket's metric cannot be self consistent, or match experimental observations.

I think this should close the case!

Professor_Fink

Zanket wrote:

You’re the pot calling the kettle black. R_ab != 0 is your claim. If you want me to do the calculation for that, then you are relying on me to do it for you. Anyone can post an equation here. Unless you show your work you have not supported your claim. (I did the calculation for my metric and it shows that R_ab = 0. See, by your own logic I just proved you wrong. So what are you complaining about?)

Actually, Zanket, I was the first to point out that R_ab was non-zero for your metric. Also, you 'proof' that R_ab is zero for your metric is a joke. It's not even close to being right.

You asked earlier if all functions of two metrics must be different if the metrics are different (I believe this was a large part of you 'proof'). The answer is no! d/dx (x + 4) and d/dx (x+100012) are both 1.

Sorry!

P.S. Logic has essentially 3 axioms:
1: The law of identity: A if and only if A
2: The law of the excluded middle: Either A or not-A
3: The law of non-contradiction: Not A and not-A
Your proof isn't logic, it's hand waving.

planck2

Zanket wrote:

You’re the pot calling the kettle black. R_ab != 0 is your claim. If you want me to do the calculation for that, then you are relying on me to do it for you. Anyone can post an equation here. Unless you show your work you have not supported your claim. (I did the calculation for my metric and it shows that R_ab = 0. See, by your own logic I just proved you wrong. So what are you complaining about?

Your insistence that logic alone cannot prove anything shows that you are not willing to listen to a well-reasoned argument.

The difference between you and me is this. I can calculate R_ab for and metric given to me, whether I have the time to do so is another matter.

I am willing to listen to well reasoned arguement, but your argument doesn't belong to that equivalence class.

Son Goku

Professor_Fink wrote:

Ok, this ends now. Here is a derivation for R_00 for both Zankets metric and the Schwarzschild metric. Note that it took 71 equations to derive this one element of the tensor. I do not plan on repeating this another 15 times.

Clearly R_00 is nonzero for Zankets metric, and R_00=0 for the Schwarzschild and Minkowski metrics.

This means that Zankets metric is not conformal to flat space, which means he cannot use arguements based on special relativity with his metric and remain consistent. Additionally R_ab is always observed to be 0 when T_ab=0 otherwise special relativity would essentially be wrong. Muon decay experiments, etc., all verify special relativity, so Zanket's metric cannot be self consistent, or match experimental observations.

I think this should close the case!

I'm only on the fourth component, I think I'll wait to see what Zanket makes of this before I continue.

Now remember Zanket, there is nothing you can do to refute the fact that R_ab = 0 when T_ab = 0. If it wasn't true Special Relativity wouldn't work. There would be no regime where SR would be an approximation.
Your universe has constant curvature.

The other thing is you have never given the calculations that show certain things:

Then, for example, the new metric predicts that the relativistic orbital precession of Mercury is 42.98 arc seconds per century, the same as the Schwarzschild metric predicts.

Do it, calculate the precession of Mercury in your metric and show us that they agree.
You can't expect us to take it on faith.

Zanket

Professor_Fink wrote:

Ok, this ends now. Here is a derivation for R_00 for both Zankets metric and the Schwarzschild metric. Note that it took 71 equations to derive this one element of the tensor. I do not plan on repeating this another 15 times.

Thanks! Now that is one fine example of supporting an argument.

I am in the process of analyzing this. I won't have a lot of free time until Monday, so it'll probably be a few days until I get an answer back to you.

In the meantime, do you confirm or deny that eq. 6 has a typo? There is no such expression in the Schwarzschild metric (at least not in the version in my paper’s reference, the version I posted above), whereas eq. 38, its complement for my metric, is an expression in my metric. It seems that the plus sign in eq. 6 should be a minus sign, in concordance with the minus sign in your usage of eq. 6 in eq. 12. If it’s not a typo, please explain the inconsistency between eqs. 6 and 38, and between eqs. 6 and 12.

Also, why is the fact that eqs. 6 and 38 are to the -1 power not reflected in the usage of those equations in eqs. 12 and 44 respectively?

Actually, Zanket, I was the first to point out that R_ab was non-zero for your metric.

It fully became planck2’s claim too when he argued it too.

You asked earlier if all functions of two metrics must be different if the metrics are different (I believe this was a large part of you 'proof'). The answer is no! d/dx (x + 4) and d/dx (x+100012) are both 1.

I said (paraphrasing) that equations that compare metrics (or any other type of equation) must fully reflect the differences among the metrics.

Zanket

planck2 wrote:

The difference between you and me is this. I can calculate R_ab for and metric given to me, whether I have the time to do so is another matter.

This is yet another empty claim. How do we know that you can calculate R_ab when you don’t show your work?

I am willing to listen to well reasoned arguement, but your argument doesn't belong to that equivalence class.

If you think your empty claims are well-reasoned arguments, then you’re right.

Zanket

Son Goku wrote:

I'm only on the fourth component, I think I'll wait to see what Zanket makes of this before I continue.

If you show it, I suggest you lay it out like the Professor did, which is nicely done.

Now remember Zanket, there is nothing you can do to refute the fact that R_ab = 0 when T_ab = 0. If it wasn't true Special Relativity wouldn't work. There would be no regime where SR would be an approximation.
Your universe has constant curvature.

I’ll come back to that if I can’t refute the Professor.

Now this is how science is done! I’m glad to see that this claim has support.

The other thing is you have never given the calculations that show certain things:
Do it, calculate the precession of Mercury in your metric and show us that they agree.

Section 6 shows that results of the new metric match results of the Schwarzschild metric to all significant digits for all experimental tests of the latter to date, in which case such calculations are superfluous. Have you not read section 6?

You can't expect us to take it on faith.

I certainly don’t expect that, which is why I wrote a rigorous proof that covers every experimental test of the Schwarzschild metric to date. Read section 6, and then tell me the first statement with which you disagree, if any.

Professor_Fink

Indeed eq. 6 has a typo, but it is just in that equation. I used g_33 = (1-2GM/r)^-1 the whole way through. I've fixed the typo in the pdf.

g^33 is not the same as g_33. g_ii*g^ii =1. And so g^ii = g_ii^-1. It standard notation in differential geometry.

Professor_Fink

Zanket wrote:

This is yet another empty claim. How do we know that you can calculate R_ab when you don’t show your work?

I know for a fact that Planck2 has a first in theoretical physics and took both differential geometry and general relativity. So he can definitely calculate R_ab.

Just as it took me ages to write up the derivation for one component, I am sure that he probably does not have the time or the inclination.

Also a few days ago, Son Goku offered to write it out, and there is hardly any point in us all deriving the same thing.

planck2

Further your metric is the Schwarzschild metric with M replaced with (-M). As I have said already this is gives and a naked singularity and moreover the spacetime is unstable. See the paper by Reinaldo J Gleiser and Gustavo Dotti in Classical and Quantum Gravity 23 (2006) 5063-5077.

Zanket

Professor_Fink wrote:

Indeed eq. 6 has a typo, but it is just in that equation. I used g_33 = (1-2GM/r)^-1 the whole way through. I've fixed the typo in the pdf.
g^33 is not the same as g_33. g_ii*g^ii =1. And so g^ii = g_ii^-1. It standard notation in differential geometry.

OK, thanks for the quick response on that.

I know for a fact that Planck2 has a first in theoretical physics and took both differential geometry and general relativity. So he can definitely calculate R_ab.

Be that as it may, he has to support his claims. We don’t simply trust even Hawking. He’s free to depend on your calculations, or Son Goku’s, but he didn’t do that. Instead he wants us to believe that my paper has a problem based on his word alone.

Zanket

planck2 wrote:

Further your metric is the Schwarzschild metric with M replaced with (-M).

No, my metric is clearly the Schwarzschild metric with sqrt(1 - (R / r)) replaced with sqrt(r / (r + R)), which is how my metric is derived in section 5. If my metric was the Schwarzschild metric with M replaced with –M, then sqrt(1 - (R / r)) in the latter would be replaced with sqrt(1 - (-R / r)) = sqrt(1 + (R / r)), which is the reciprocal of sqrt(r / (r + R)).

planck2

Zanket wrote:

OK, thanks for the quick response on that.


Be that as it may, he has to support his claims. We don’t simply trust even Hawking. He’s free to depend on your calculations, or Son Goku’s, but he didn’t do that. Instead he wants us to believe that my paper has a problem based on his word alone.

i don' think so. you expect us to believe you on your word alone.
You expected us to believe R_ab was non zero for Schwarschild based on your logical arguements. As for relying on Prof Fink and Son Goku they would probably agree that I need no help from them
"experimental" verifications my eye.

as for replacing with -M. well then it seems you might be correct. However to correctly get your metric you replace M with - M and swap g_00 and g_11.

Zanket

Professor_Fink wrote:

Here is a derivation for R_00 for both Zankets metric and the Schwarzschild metric.

It looks like there is another typo, in eq. 22, copied to eq. 23. It seems that the plus sign should be a minus sign, in concordance with eqs. 14 and 24. Do you confirm?

Zanket

planck2 wrote:

i don' think so. you expect us to believe you on your word alone.
You expected us to believe R_ab was non zero for Schwarschild based on your logical arguements.

A logical argument is obviously not my word alone.

As for relying on Prof Fink and Son Goku they would probably agree that I need no help from them

That’s beside the point.

"experimental" verifications my eye.

If only theories could be refuted that way.

However to correctly get your metric you replace M with - M and swap g_00 and g_11.

I don’t confirm or deny that (too lazy to verify in the absence of an identified problem). Do you see a problem with that?

Professor_Fink

Yes eqs 22 and 23 have typos. Essentially I made the mistake with g_33 the whole way through, and then noticed my mistake, so I went back through the paper fixing the equations before I posted the original, but it seems I have left a few typos uncorrected. As you can see, though, it is correct again from eq 24 onwards. I've fixed the typo in the PDF.

Zanket

Professor_Fink wrote:

Here is a derivation for R_00 for both Zankets metric and the Schwarzschild metric.
...
Clearly R_00 is nonzero for Zankets metric, and R_00=0 for the Schwarzschild and Minkowski metrics.

I’ve analyzed your derivations. (Thanks again for writing them up.) First, based on the assumption that your derivation for the Schwarzschild metric is correct (I see no obvious mathematical mistakes), I give here a rigorous proof that R_00 for my metric too:

The only inputs to your derivation for the Schwarzschild metric are g_00, g_11, g_22, and g_33. You said:

g^33 is not the same as g_33. g_ii*g^ii =1. And so g^ii = g_ii^-1. It standard notation in differential geometry.

Then it is clear from eyeballing your derivation that g_11 and g_22 always drop out. That leaves only g_00 and g_33 as inputs, both of which refer to eq. 8 in my paper for the Schwarzschild metric:

Eq. 8 = 1 / gamma = sqrt(1 – (R / r))

My metric is derived by just replacing instances of eq. 8 with eq. 9:

Eq. 9 = 1 / gamma = sqrt(r / (r + R)) = sqrt(1 – (R / (r + R)))

You can see that the only difference between eqs. 8 and 9—hence the only difference between g_00 and g_33 for the metrics in your derivations—is that the sole r in eq. 8 is replaced with r + R in eq. 9. Then R_00 for my metric is calculable by plugging in the value (not the symbols) r + R for r into g_00 and g_33 for your derivation for the Schwarzschild metric, which always returns R_00 = 0. (For example, if R = 1 and r = 2 for the Schwarzschild metric, then plug in R = 1 and r = (2 + 1) = 3 to get R_00 for my metric for the inputs R = 1 and r = 2.) Then it is a mathematical certainty that R_00 = 0 for my metric too (again, assuming that your derivation for the Schwarzschild metric is correct). This completes my proof.

The proof tells me that there is at least one error in your analysis for my metric. To find them, I carefully entered your equations for each metric side by side in a spreadsheet, using sample inputs (e.g. R = 1 and r = 2). I verified the equations using the cross-reference equations (equivalent equations) in your analysis, and confirmed that R_00 = 0 for the Schwarzschild metric. Then I copied into a third column your equations for the Schwarzschild metric, except that I changed the value (the input value, not the symbol) for r, to r + R. (For example, if R = 1 and r = 2 for my metric, then in the third column I used R = 1 and r = (2 + 1) = 3.) Results in the column in my spreadsheet for my metric that do not match the corresponding result in the third column is either an equation in error, or based on an equation in error.

There are errors in eqs. 50, 55, and 63. I analyzed those equations further. This is what I found:

Eq. 50: The expression ∂/(∂x^3) should be equivalent to the expression in ∂/(∂x^3) in eq. 18 except that r is replaced with r + R, but it’s not. Instead it is equivalent to twice the expression ∂/(∂x^3) in eq. 18, for no apparent justifiable reason.

Eq. 55: The second line should be equivalent to the second line in eq. 23 except that r is replaced with r + R, but it’s not. Instead it is equivalent to the second line in eq. 23, for no apparent justifiable reason.

Eq. 63: The expression ∂/(∂x^3) is equivalent to R / (r^2 * g_00), the same as the expression ∂/(∂x^3) in eq. 31, for no apparent justifiable reason. The r not in g_00 in eq. 63 should be replaced with r + R.

When these errors are corrected and the corrections propagated as appropriate, R_00 = 0 for my metric.

Can you refute that?

Son Goku

You can see that the only difference between eqs. 8 and 9—hence the only difference between g_00 and g_33 for the metrics in your derivations—is that the sole r in eq. 8 is replaced with r + R in eq. 9. Then R_00 for my metric is calculable by plugging in the value (not the symbols) r + R for r into g_00 and g_33 for your derivation for the Schwarzschild metric, which always returns R_00 = 0. (For example, if R = 1 and r = 2 for the Schwarzschild metric, then plug in R = 1 and r = (2 + 1) = 3 to get R_00 for my metric for the inputs R = 1 and r = 2.) Then it is a mathematical certainty that R_00 = 0 for my metric too (again, assuming that your derivation for the Schwarzschild metric is correct). This completes my proof.

No Zanket, you have to completely work out the Ricci Tensor again for a different metric. At no point are you allowed to convert r to r + R except at the very start, before the derivation. The whole process is nonlinear so you can't do this.

Eq. 50: The expression ∂/(∂x^3) should be equivalent to the expression in ∂/(∂x^3) in eq. 18 except that r is replaced with r + R, but it’s not. Instead it is equivalent to twice the expression ∂/(∂x^3) in eq. 18, for no apparent justifiable reason.

It shouldn't Zanket, work out the differential in both cases. From Eq. 15 in the first case and Eq. 47 in the second case. You'll see that they result in very different answers.
I can show you if you want.

Eq. 55: The second line should be equivalent to the second line in eq. 23 except that r is replaced with r + R, but it’s not. Instead it is equivalent to the second line in eq. 23, for no apparent justifiable reason.

Look at the two Christoffel symbols being multiplied. If you work out the derivation you'll see they can't come out the same.
I can show you if you want.

Eq. 63: The expression ∂/(∂x^3) is equivalent to R / (r^2 * g_00), the same as the expression ∂/(∂x^3) in eq. 31, for no apparent justifiable reason. The r not in g_00 in eq. 63 should be replaced with r + R.

Zanket do you understand these equations?
I just worked that out and got the answer Professor_Fink got.
What do you think ∂/(∂x^3) means?
Please answer this question because it is important in order to show you how the derivation works.

Remember the difference between raised and lowered components involves letting the metric or inverse metric act on one to give the other.

i.e., x^i = g^ik x_k

Professor_Fink

Zanket wrote:

You can see that the only difference between eqs. 8 and 9—hence the only difference between g_00 and g_33 for the metrics in your derivations—is that the sole r in eq. 8 is replaced with r + R in eq. 9. Then R_00 for my metric is calculable by plugging in the value (not the symbols) r + R for r into g_00 and g_33 for your derivation for the Schwarzschild metric, which always returns R_00 = 0. (For example, if R = 1 and r = 2 for the Schwarzschild metric, then plug in R = 1 and r = (2 + 1) = 3 to get R_00 for my metric for the inputs R = 1 and r = 2.) Then it is a mathematical certainty that R_00 = 0 for my metric too (again, assuming that your derivation for the Schwarzschild metric is correct). This completes my proof.

No, you are making a mistake here. The problem is that x^4 = r. If we replace r+R with r in you metric, then we also have to change x^4 to r+R. You're only doing the coordinate transform on the metric but not on x. This is why we are getting different answers. You cannot ignore x^4=r.

Surely you can see that my derivation is correct (up to a possible typo). There is no arguement you can put forth to show it's wrong, unless you spot a mistake in the mathematics.

Professor_Fink

Zanket wrote:

There are errors in eqs. 50, 55, and 63. I analyzed those equations further. This is what I found:

Eq. 50: The expression ∂/(∂x^3) should be equivalent to the expression in ∂/(∂x^3) in eq. 18 except that r is replaced with r + R, but it’s not. Instead it is equivalent to twice the expression ∂/(∂x^3) in eq. 18, for no apparent justifiable reason.

Eq. 55: The second line should be equivalent to the second line in eq. 23 except that r is replaced with r + R, but it’s not. Instead it is equivalent to the second line in eq. 23, for no apparent justifiable reason.

Eq. 63: The expression ∂/(∂x^3) is equivalent to R / (r^2 * g_00), the same as the expression ∂/(∂x^3) in eq. 31, for no apparent justifiable reason. The r not in g_00 in eq. 63 should be replaced with r + R.

When these errors are corrected and the corrections propagated as appropriate, R_00 = 0 for my metric.

Can you refute that?

Yes, of course I can refute this. Those quantities are christoffel symbols (the capital gammas) and so already contain partial derivatives with respect to r, so of course the there is differences between the two sections, d/dr g_00 and d/dr g_33 are not the same for both metrics!

planck2

Didn't I say he could not calculate

Zanket

Son Goku wrote:

No Zanket, you have to completely work out the Ricci Tensor again for a different metric. At no point are you allowed to convert r to r + R except at the very start, before the derivation. The whole process is nonlinear so you can't do this.

In my proof, all I do is convert r to r + R at the very start, before the derivation. You agree that I can do that, so the proof holds.

Now I elaborate on my proof. Keep in mind that it involves only the Professor’s derivation for the Schwarzschild metric; it does not involve the derivation for my metric. In the Professor’s derivation for the Schwarzschild metric, consider:

g_00 = -1 + (R / r)
g_33 = (1 – (R / r))^1

Let y = r. (Nothing amiss there; it’s just algebraic substitution, which is always allowed.) Then for the Schwarzschild metric:

g_00 = -1 + (R / y)
g_33 = (1 – (R / y))^1

In the derivation, g_11 and g_22 drop out, and there are no other inputs. So had the Professor done the derivation using the versions of g_00 and g_33 having r replaced with y, the result would have been the same, namely R_00 = 0. And nothing prevents him from being able to do that. Do you agree? (Please answer that.)

For my metric let y = r + R. (Nothing amiss there; it’s just algebraic substitution, which is always allowed.) Then for my metric:

g_00 = -1 + (R / y)
g_33 = (1 – (R / y))^1

Notice that these inputs are identical to those for the Schwarzschild metric immediately above. Then all the inputs are the same for both metrics, which means that the derivation for R_00 for the Schwarzschild metric can substitute for the derivation for my metric. Nothing prevents me, when the derivation is done, from plugging in a value of r + R for y, instead of r, to calculate a value for R_00. Then it is a mathematical certainty that R_00 = 0 for my metric too, assuming that the Professor’s derivation for the Schwarzschild metric is valid. Do you agree? (Please answer that.)

Zanket wrote:

Eq. 50: The expression ∂/(∂x^3) should be equivalent to the expression in ∂/(∂x^3) in eq. 18 except that r is replaced with r + R, but it’s not. Instead it is equivalent to twice the expression ∂/(∂x^3) in eq. 18, for no apparent justifiable reason.

It shouldn't Zanket, work out the differential in both cases. From Eq. 15 in the first case and Eq. 47 in the second case. You'll see that they result in very different answers.
I can show you if you want.

I agree that they should differ; in what you quoted of mine, I said they should differ (“should be equivalent except that ...”). What I disagree with is how they differ. There’s no justification for one being twice the other. But yes, please show me the work. I’d like to see it.

Zanket wrote:

Eq. 55: The second line should be equivalent to the second line in eq. 23 except that r is replaced with r + R, but it’s not. Instead it is equivalent to the second line in eq. 23, for no apparent justifiable reason.

Look at the two Christoffel symbols being multiplied. If you work out the derivation you'll see they can't come out the same.
I can show you if you want.

You misread my quote. I said what you’re saying; namely, the results should be different (“should be equivalent except that ...”). But the expressions for the second line in eqs. 23 and 55 do not differ in the Professor's work; both are equivalent to 4 / r. Then you agree with me, in which case you implicitly agree that the Professor's derivation is invalid. But yes, please show me the work.

What do you think ∂/(∂x^3) means?
Please answer this question because it is important in order to show you how the derivation works.

I don’t fully know, but it became clear in my analysis that fully knowing the meaning of ∂/(∂x^3) is unnecessary to make my points. What I do know is that they are equivalent to another expression containing r, and it seems that this is all I need to know.

For example, it is easy to see that, given eqs. 25 and 57, the second line in both eqs. 23 and 55 is equivalent to 4 / r. You agreed with me that these should differ, so you implicitly agree that the Professor’s derivation is invalid.

Son Goku

Zanket wrote:

You agreed with me that these should differ, so you implicitly agree that the Professor’s derivation is invalid.

Hilarious Zanket, you know that's not what I'm saying.

I don’t fully know, but it became clear in my analysis that fully knowing the meaning of ∂/(∂x^3) is unnecessary to make my points.

It is completely necessary. Saying that it isn't doesn't make any sense.
Every calculation in that pdf involves something of the form ∂/(∂x^i), you need to know how it works.

To be more simplistic look at the derivative with respect to x_3 of your g_00 and the Schwarschild g_00 and you will see how the answers are very different.
You're using linear logic from elementary algebra on a nonlinear tensor calculation.

I'll respond to your other points later unless somebody else does first.

As Professor_Fink said ∂/(∂x^i) is built into most of the Tensorial quantities.

Professor_Fink

Zanket wrote:

In my proof, all I do is convert r to r + R at the very start, before the derivation. You agree that I can do that, so the proof holds.

Now I elaborate on my proof. Keep in mind that it involves only the Professor’s derivation for the Schwarzschild metric; it does not involve the derivation for my metric. In the Professor’s derivation for the Schwarzschild metric, consider:

g_00 = -1 + (R / r)
g_33 = (1 – (R / r))^1

Let y = r. (Nothing amiss there; it’s just algebraic substitution, which is always allowed.) Then for the Schwarzschild metric:

g_00 = -1 + (R / y)
g_33 = (1 – (R / y))^1

In the derivation, g_11 and g_22 drop out, and there are no other inputs. So had the Professor done the derivation using the versions of g_00 and g_33 having r replaced with y, the result would have been the same, namely R_00 = 0. And nothing prevents him from being able to do that. Do you agree? (Please answer that.)

I'm afraid you are wrong. x^3 = r, as I have mentioned in my derivation. You need to change x^3 to y-R. You're just using y. That is incorrect. You're not using the substitution consistently. So, no, I do not agree with you, and I have proved, rigorously, that R_00!=0. My derivation is accurate, and you seem not to understand how calculus works.

Zanket wrote:

I don’t fully know, but it became clear in my analysis that fully knowing the meaning of ∂/(∂x^3) is unnecessary to make my points. What I do know is that they are equivalent to another expression containing r, and it seems that this is all I need to know.

For example, it is easy to see that, given eqs. 25 and 57, the second line in both eqs. 23 and 55 is equivalent to 4 / r. You agreed with me that these should differ, so you implicitly agree that the Professor’s derivation is invalid.

Aaaaaaaaarrrrrrrrggghhh! ∂/(∂x^3) = ∂/∂r which is the partial derivative with respect to r. Your claim that R_00=0 is completely wrong, as you are not changing this r when you make the substitution. You NEED to do this.